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Positive-definite function

for all choices , . The set of positive-definite functions on forms a cone in the space of all bounded functions on which is closed with respect to the operations of multiplication and complex conjugation.

The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. Positive functional) on the group algebra and unitary representations of the group (cf. Unitary representation). More precisely, let be any function and let be the functional given by

then for to be positive it is necessary and sufficient that be a positive-definite function. Further, defines a -representation of the algebra on a Hilbert space , and therefore a unitary representation of the group , where for some . Conversely, for any representation and any vector , the function is a positive-definite function.

If is a topological group, the representation is weakly continuous if and only if the positive-definite function is continuous. If is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on .

For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function on a compact group is a positive-definite function if and only if its Fourier transform takes positive (operator) values on each element of the dual object, i.e.

for any representation and any vector , where is the space of .

References

Comments

The representations of associated to positive functionals mentioned above are cyclic representations. A cyclic representation of a -algebra is a representation , the -algebra of bounded operators on the Hilbert space , such that there is a vector such that the closure of is all of . These are the basic components of any representation. Indeed, if is non-degenerate, i.e. , then is a direct sum of cyclic representations. Cf. also Cyclic module for an analogous concept in ring and module theory.

The cyclic representation associated to a positive functional on is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on by

and define a left ideal of by

The inner product just defined descends to define an inner product on the quotient space . Now complete this space to obtain a Hilbert space , and define the representation by:

where denotes the class of in . The operator extends to a bounded operator on .

If contains an identity, then the class of that identity is a cyclic vector for . If does not contain an identity, such is first adjoined to obtain a -algebra and the construction is repeated for . To prove that then the class of 1 is cyclic for (not just ) one uses an approximate identity for , i.e. a net (directed set) of positive elements such that , implies and for all . Such approximate identities always exist. See e.g. [1], vol. 1, p. 321 and [a5], Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this.

A positive functional on a -algebra of norm 1 is often called a state, especially in the theoretical physics literature.